Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.

Call number $A$ as interesting if $A$ is divided by every number that can be received from $A$ by crossing some last digits. Find maximum interesting number with different digits.

Let $P(x)$ be a monic polynomial with integer coefficients of degree $10$. Prove that there exist distinct positive integers $a,b$ not exceeding $101$ such that $P(a)-P(b)$ is divisible by $101$.

Let $f:\mathbb N\to (0,\infty)$ satisfy $\prod_{d\mid n} f(d) = 1$ for every $n$ which is not prime. Determine the maximum possible number of $n$ with $1\le n \le 100$ and $f(n)\ne 1$.

Find all positive integers $m, n$ such that $\frac{2m-1}{n}$ and $\frac{2n-1}{m}$ are both integers.

The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$

Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers: $a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.

For two positive integers a and b, which are relatively prime, find all integer that can be the great common divisor of $a+b$ and $\frac{a^{2005}+b^{2005}}{a+b}$.

Does there exist a prime number whose decimal representation is of the form $3811\cdots11$ (that is, consisting of the digits $3$ and $8$ in that order, followed by one or more digits $1$)?

[u]Round 1[/u] [b]p1.[/b] Suppose that gold satisfies the relation $p = v + v^2$, where $p$ is the price and $v$ is the volume. How many pieces of gold with volume $1$ can be bought for the price of a piece with volume $2$? [b]p2.[/b] Find the smallest prime number with each digit greater or equal to $8$. [b]p3.[/b] What fraction of regular hexagon $ZUMING$ is covered by both quadrilateral $ZUMI$ and quadrilateral$ MING$? [u]Round 2[/u] [b]p4.[/b] The two smallest positive integers expressible as the sum of two (not necessarily positive) perfect cubes are $1 = 1^3 +0^3$ and $2 = 1^3 +1^3$. Find the next smallest positive integer expressible in this form. [b]p5.[/b] In how many ways can the numbers $1, 2, 3,$ and $4$ be written in a row such that no two adjacent numbers differ by exactly $1$? [b]p6.[/b] A real number is placed in each cell of a grid with $3$ rows and $4$ columns. The average of the numbers in each column is $2016$, and the average of the numbers in each row is a constant $x$. Compute $x$. [u]Round 3[/u] [b]p7.[/b] Fardin is walking from his home to his oce at a speed of $1$ meter per second, expecting to arrive exactly on time. When he is halfway there, he realizes that he forgot to bring his pocketwatch, so he runs back to his house at $2$ meters per second. If he now decides to travel from his home to his office at $x$ meters per second, find the minimum $x$ that will allow him to be on time. [b]p8.[/b] In triangle $ABC$, the angle bisector of $\angle B$ intersects the perpendicular bisector of $AB$ at point $P$ on segment $AC$. Given that $\angle C = 60^o$, determine the measure of $\angle CPB$ in degrees. [b]p9.[/b] Katie colors each of the cells of a $6\times 6$ grid either black or white. From top to bottom, the number of black squares in each row are $1$, $2$, $3$, $4$, $5$, and $6$, respectively. From left to right, the number of black squares in each column are $6$, $5$, $4$, $3$, $2$, and $1$, respectively. In how many ways could Katie have colored the grid? [u]Round 4[/u] [b]p10.[/b] Lily stands at the origin of a number line. Each second, she either moves $2$ units to the right or $1$ unit to the left. At how many different places could she be after $2016$ seconds? [b]p11.[/b] There are $125$ politicians standing in a row. Each either always tells the truth or always lies. Furthermore, each politician (except the leftmost politician) claims that at least half of the people to his left always lie. Find the number of politicians that always lie. [b]p12.[/b] Two concentric circles with radii $2$ and $5$ are drawn on the plane. What is the side length of the largest square whose area is contained entirely by the region between the two circles? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2934055p26256296]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes. \[ 2^m p^2 + 1 = q^5 \]

Five distinct positive integers form an arithmetic progression. Can their product be equal to $a^{2008}$ for some positive integer $a$ ?

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

Find (with proof) all natural numbers $n$ such that, for some natural numbers $a$ and $b$, $a\ne b$, the digits in the decimal representations of the two numbers $n^a+1$ and $n^b+1$ are in reverse order.

Suppose the polynomial $f(x) = x^{2014}$ is equal to $f(x) =\sum^{2014}_{k=0} a_k {x \choose k}$ for some real numbers $a_0,... , a_{2014}$. Find the largest integer $m$ such that $2^m$ divides $a_{2013}$.

Bob is making partitions of $10$, but he hates even numbers, so he splits $10$ up in a special way. He starts with $10$, and at each step he takes every even number in the partition and replaces it with a random pair of two smaller positive integers that sum to that even integer. For example, $6$ could be replaced with $1+5$, $2+4$, or $3+3$ all with equal probability. He terminates this process when all the numbers in his list are odd. The expected number of integers in his list at the end can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Michael Ren[/i]

For any given prime $p$, determine whether the equation $x^2 + y^2 + p^z = 2003$ always has integer solutions in $x, y, z$. Justify your answer

[i]Each problem in this section will depend on the previous one! The values $A, B, C$, and $D$ refer to the answers to problems $1, 2, 3$, and $4$, respectively.[/i] [b]TR1.[/b] The number $2020$ has three different prime factors. What is their sum? [b]TR2.[/b] Let $A$ be the answer to the previous problem. Suppose$ ABC$ is a triangle with $AB = 81$, $BC = A$, and $\angle ABC = 90^o$. Let $D$ be the midpoint of $BC$. The perimeter of $\vartriangle CAD$ can be written as $x + y\sqrt{z}$, where $x, y$, and $z$ are positive integers and $z$ is not divisible by the square of any prime. What is $x + y$? [b]TR3.[/b] Let $B$ the answer to the previous problem. What is the unique real value of $k$ such that the parabola $y = Bx^2 + k$ and the line $y = kx + B$ are tangent? [b]TR4.[/b] Let $C$ be the answer to the previous problem. How many ordered triples of positive integers $(a, b, c)$ are there such that $gcd(a, b) = gcd(b, c) = 1$ and $abc = C$? [b]TR5.[/b] Let $D$ be the answer to the previous problem. Let $ABCD$ be a square with side length $D$ and circumcircle $\omega$. Denote points $C'$ and $D'$ as the reflections over line $AB$ of $C$ and $D$ respectively. Let $P$ and $Q$ be the points on $\omega$, with$ A$ and $P$ on opposite sides of line $BC$ and $B$ and $Q$ on opposite sides of line $AD$, such that lines $C'P$ and $D'Q$ are both tangent to $\omega$. If the lines $AP$ and $BQ$ intersect at $T$, what is the area of $\vartriangle CDT$? PS. You had better use hide for answers.

[b]p1.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. His drink contains $45\%$ of orange juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $60\%$ of orange juice? [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] For one particular number $a > 0$ the function f satisfies the equality $f(x + a) =\frac{1 + f(x)}{1 - f(x)}$ for all $x$. Show that $f$ is a periodic function. (A function $f$ is periodic with the period $T$ if $f(x + T) = f(x)$ for any $x$.) [b]p4.[/b] If $a, b, c, x, y, z$ are numbers so that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}= 1$ and $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}= 0$. Show that $\frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} = 1$ [b]p5.[/b] Is it possible that a four-digit number $AABB$ is a perfect square? (Same letters denote the same digits). [b]p6.[/b] A finite number of arcs of a circle are painted black (see figure). The total length of these arcs is less than $\frac15$ of the circumference. Show that it is possible to inscribe a square in the circle so that all vertices of the square are in the unpainted portion of the circle. [img]https://cdn.artofproblemsolving.com/attachments/2/c/bdfa61917a47f3de5dd3684627792a9ebf05d5.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a} <1$$ where $a, b$ and $c$ are positive numbers such that $abc = 1$. (G Galperin)

Given an integer $n\ge 2$, show that there exist two integers $a,b$ which satisfy the following. For all integer $m$, $m^3+am+b$ is not a multiple of $n$.

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

For each positive integer $n$, let $s_n$ be the number of permutations $(a_1, a_2, \cdots, a_n)$ of $(1, 2, \cdots, n)$ such that $\dfrac{a_1}{1} + \dfrac{a_2}{2} + \cdots + \dfrac{a_n}{n}$ is a positive integer. Prove that $s_{2n} \ge n$ for all positive integer $n$.

Use the following description of a machine to solve the first 4 problems in the round. A machine displays four digits: $0000$. There are two buttons: button $A$ moves all digits one position to the left and fills the rightmost position with $0$ (for example, it changes $1234$ to $2340$), and button $B$ adds $11$ to the current number, displaying only the last four digits if the sum is greater than $9999$ (for example, it changes $1234$ to $1245$, and changes $9998$ to $0009$). We can denote a sequence of moves by writing down the buttons pushed from left to right. A sequence of moves that outputs $2100$, for example, is $BABAA$. [b]p1[/b]. Give a sequence of $17$ or less moves so that the machine displays $2020$. [b]p2.[/b] Using the same machine, how many outputs are possible if you make at most three moves? [b]p3.[/b] Button $ B$ now adds n to the four digit display, while button $ A$ remains the same. For how many positive integers $n \le 20$ (including $11$) can every possible four-digit output be reached? [b]p4.[/b] Suppose the function of button $ A$ changes to: move all digits one position to the right and fill the leftmost position with $2$. Then, what is the minimum number of moves required for the machine to display $2020$, if it initially displays $0000$? [b]p5.[/b] In the figure below, every inscribed triangle has vertices that are on the midpoints of its circumscribed triangle’s sides. If the area of the largest triangle is $64$, what is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/6/f/fe17b6a6d0037163f0980a5a5297c1493cc5bb.png[/img] [b]p6.[/b] A bee flies $10\sqrt2$ meters in the direction $45^o$ clockwise of North (that is, in the NE direction). Then, the bee turns $135^o$ clockwise, and flies $20$ forward meters. It continues by turning $60^o$ counterclockwise, and flies forward $14$ meters. Finally, the bee turns $120^o$ clockwise and flies another $14$ meters forward before finally finding a flower to pollinate. How far is the bee from its starting location in meters? [b]p7.[/b] All the digits of a $15$-digit number are either $p$ or $c$. $p$ shows up $3$ more times than $c$ does, and the average of the digits is $c - p$. What is $p + c$? [b]p8.[/b] Let $m$ be the sum of the factors of $75$ (including $1$ and $75$ itself). What is the ones digit of $m^{75}$ ? [b]p9.[/b] John flips a coin twice. For each flip, if it lands tails, he does nothing. If it lands heads, he rolls a fair $4$-sided die with sides labeled 1 through $4$. Let $a/b$ be the probability of never rolling a $3$, in simplest terms. What is $a + b$? [b]p10.[/b] Let $\vartriangle ABC$ have coordinates $(0, 0)$, $(0, 3)$,$(18, 0)$. Find the number of integer coordinates interior (excluding the vertices and edges) of the triangle. [b]p11.[/b] What is the greatest integer $k$ such that $2^k$ divides the value $20! \times 20^{20}$? [b]p12.[/b] David has $n$ pennies, where $n$ is a natural number. One apple costs $3$ pennies, one banana costs $5$ pennies, and one cranberry costs $7$ pennies. If David spends all his money on apples, he will have $2$ pennies left; if David spends all his money on bananas, he will have $4$ pennies left; is David spends all his money on cranberries, he will have $6$ pennies left. What is the second least possible amount of pennies that David can have? [b]p13.[/b] Elvin is currently at Hopperville which is $40$ miles from Waltimore and $50$ miles from Boshington DC. He takes a taxi back to Waltimore, but unfortunately the taxi gets lost. Elvin now finds himself at Kinsville, but he notices that he is still $40$ miles from Waltimore and $50$ miles from Boshington $DC$. If Waltimore and Boshington DC are $30$ miles apart, What is the maximum possible distance between Hopperville and Kinsville? [b]p14.[/b] After dinner, Rick asks his father for $1000$ scoops of ice cream as dessert. Rick’s father responds, “I will give you $2$ scoops of ice cream, plus $ 1$ additional scoop for every ordered pair $(a, b)$ of real numbers satisfying $\frac{1}{a + b}= \frac{1}{a}+ \frac{1}{b}$ you can find.” If Rick finds every solution to the equation, how many scoops of ice cream will he receive? [b]p15.[/b] Esther decides to hold a rock-paper-scissors tournament for the $56$ students at her school. As a rule, competitors must lose twice before they are eliminated. Each round, all remaining competitors are matched together in best-of-1 rock-paper-scissors duels. If there is an odd number of competitors in a round, one random competitor will not compete that round. What is the maximum number of matches needed to determine the rock-paper-scissors champion? [b]p16.[/b] $ABCD$ is a rectangle. $X$ is a point on $\overline{AD}$, $Y$ is a point on $\overline{AB}$, and $N$ is a point outside $ABCD$ such that $XYNC$ is also a rectangle and $YN$ intersects $\overline{BC}$ at its midpoint $M$. $ \angle BYM = 45^o$. If $MN = 5$, what is the sum of the areas of $ABCD$ and $XYNC$? [b]p17. [/b] Mr. Brown has $10$ identical chocolate donuts and $15$ identical glazed donuts. He knows that Amar wants $6$ donuts, Benny wants $9$ donuts, and Callie wants $9$ donuts. How many ways can he distribute out his $25$ donuts? [b]p18.[/b] When Eric gets on the bus home, he notices his $ 12$-hour watch reads $03: 30$, but it isn’t working as expected. The second hand makes a full rotation in $4$ seconds, then makes another in $8$ seconds, then another in $ 12$ seconds, and so on until it makes a full rotation in $60$ seconds. Then it repeats this process, and again makes a full rotation in $4$ second, then $8$ seconds, etc. Meanwhile, the minute hand and hour hand continue to function as if every full rotation of the second hand represents $60$ seconds. When Eric gets off the bus $75$ minutes later, his watch reads $AB: CD$. What is $A + B + C + D$? [b]p19.[/b] Alex and Betty want to meet each other at the airport. Alex will arrive at the airport between $12: 00$ and $13: 15$, and will wait for Betty for $15$ minutes before he leaves. Betty will arrive at the airport between $12: 30$ and $13: 10$, and will wait for Alex for $10$ minutes before she leaves. The chance that they arrive at any time in their respective time intervals is equally likely. The probability that they will meet at the airport can be expressed as $a/b$ where $a/b$ is a fraction written in simplest form. What is $a + b$? [b]p20.[/b] Let there be $\vartriangle ABC$ such that $A = (0, 0)$, $B = (23, 0)$, $C = (a, b)$. Furthermore, $D$, the center of the circle that circumscribes $\vartriangle ABC$, lies on $\overline{AB}$. Let $\angle CDB = 150^o$. If the area of $\vartriangle ABC$ is $m/n$ where $m, n$ are in simplest integer form, find the value of $m \,\, \mod \,\,n$ (The remainder of $m$ divided by $n$). PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For integers $a$ and $b$ with $0 \leq a,b < {2010}^{18}$ let $S$ be the set of all polynomials in the form of $P(x)=ax^2+bx.$ For a polynomial $P$ in $S,$ if for all integers n with $0 \leq n <{2010}^{18}$ there exists a polynomial $Q$ in $S$ satisfying $Q(P(n)) \equiv n \pmod {2010^{18}},$ then we call $P$ as a [i]good polynomial.[/i] Find the number of [i]good polynomials.[/i]